In this article, we prove a restriction theorem for the Fourier–Hermite transform and obtain Strichartz estimate system of orthonormal functions Hermite operator $$H=-\Delta +|x|^2$$ on $${\mathbb {R}}^n$$ as application. Further, show optimal behavior constant in limit large number functions.

In this paper we develop a general and very simple construction for complete orthonormal bases for system identiication. This construction provides a unifying formulation of all known orthonormal bases since the common FIR and recently popular Laguerre and Kautz model structures are restrictive special cases of our construction as is another construction method based on balanced realisations of...

The supercell- based orthonormal basis method is proposed to investigate the modal properties of the Bragg fibers. A square lattice is constructed by the whole Bragg fiber which is considered as a supercell, and the periodical dielectric structure of the square lattice is decomposed using periodic functions (cosine). The modal electric field is expanded as the sum of the orthonormal set of Herm...

Spread-spectrum watermarking, in which random noise is added to transform coefficients and detected with a correlation operator, has become a preferred paradigm for many watermarking applications. This paper analyzes the performance of such a watermarking system when the underlying transform is a tight frame rather than a traditional orthonormal expansion. The analysis indicates that a tight fr...

In the present work a newer type of black box nonlinear model in Hammerstein structure is proposed. The model has Wavelet Network coupled with Orthonormal Basis Functions which is capable of modeling a class of non-linear systems with acceptable accuracy. Wavelet basis functions have the property of localization in both the time and frequency domains which enables wavelet networks to approximat...

Regression or function classes of Euclidean type with compact support and certain smoothness properties are shown to be PAC learnable by the Nadaraya-Watson estimator based on complete orthonormal systems. While requiring more smoothness properties than typical PAC formulations, this estimator is computationally efficient, easy to implement, and known to perform well in a number of practical ap...

The last decade has seen a resurgence of interest in the use of ortho-normalised parameterisations for system estimation applications. This has produced a large body of theoretical work that has sought to provide sound scientific underpinnings. The paper here has a different focus in that emphasis is placed on practical dividends associated with orthonormal parameterisations.

Many nonlinear systems can be described by a Wiener-Schetzen model. In this model, the linear dynamics are formulated in terms of orthonormal basis functions (OBFs). The nonlinearity is modeled by a multivariate polynomial. In general, an infinite number of OBFs is needed for an exact representation of the system. This paper considers the approximation of a Wiener system with finite-order infin...

There has recently been interest in the use of orthonor-mal bases for the purposes of SISO system identiication. Concurrently, but separately, there has also been vigorous work on estimation of MIMO systems by computa-tionally cheap and reliable schemes. These latter ideas have collectively become known as`4SID' methods. This paper is a contribution overlapping these two schools of thought by s...

The purpose of this paper is threefold. Firstly, it is to establish that contrary to what might be expected, the accuracy of well known and frequently used asymptotic variance results can depend on choices of fixed poles or zeros in the model structure. Secondly, it is to derive new variance expressions that can provide greatly improved accuracy while also making explicit the influence of any f...